Square Roots

To solve $\sqrt{\sqrt{2}}$, we will use the property of roots.

• Step 1: Recognize that $\sqrt{\sqrt{2}}$ involves two square roots.
• Step 2: Each square root can be expressed using exponents: $\sqrt{2} = 2^{1/2}$.
• Step 3: Therefore, $\sqrt{\sqrt{2}} = (2^{1/2})^{1/2}$.
• Step 4: Apply the formula for the root of a root: $(x^{a})^{b} = x^{ab}$.
• Step 5: For $(2^{1/2})^{1/2}$, this means we compute the product of the exponents: $(1/2) \times (1/2) = 1/4$.
• Step 6: The expression simplifies to $2^{1/4}$, which is written as $\sqrt[4]{2}$.

Therefore, $\sqrt{\sqrt{2}} = \sqrt[4]{2}$.

This corresponds to choice 2: $\sqrt[4]{2}$.

The solution to the problem is $\sqrt[4]{2}$.

To solve the problem of finding $\sqrt[5]{\sqrt[3]{5}}$, we'll use the formula for a root of a root, which combines the exponents:

• Step 1: Express each root as an exponent.
  We start with the innermost root: $\sqrt[3]{5} = 5^{1/3}$.
• Step 2: Apply the outer root.
  The square root to the fifth power is expressed as: $\sqrt[5]{5^{1/3}} = (5^{1/3})^{1/5}$.
• Step 3: Combine the exponents.
  Using the exponent rule $(a^m)^n = a^{m \times n}$, we get:
  $(5^{1/3})^{1/5} = 5^{(1/3) \times (1/5)} = 5^{1/15}$.
• Step 4: Convert the exponent back to root form.
  This can be written as $\sqrt[15]{5}$.

Therefore, the simplified expression of $\sqrt[5]{\sqrt[3]{5}}$ is $\sqrt[15]{5}$.

To solve this problem, we'll observe the following process:

• Step 1: Recognize the expression $\sqrt[10]{\sqrt[10]{1}}$ involves nested roots.
• Step 2: Apply the formula for nested roots: $\sqrt[n]{\sqrt[m]{x}} = \sqrt[n \cdot m]{x}$.
• Step 3: Set $n = 10$ and $m = 10$, resulting in $\sqrt[10 \times 10]{1} = \sqrt[100]{1}$.
• Step 4: Simplify $\sqrt[100]{1}$. Any root of 1 is 1, as $1^k = 1$ for any positive rational number $k$.

Thus, the evaluation of the original expression $\sqrt[10]{\sqrt[10]{1}}$ equals 1.

Comparing this result to the provided choices:

• Choice 1 is $1$.
• Choice 2 is $\sqrt[100]{1}$, which is also 1.
• Choice 3 is $\sqrt{1} = 1$.
• Choice 4 states all answers are correct.

Therefore, choice 4 is correct: All answers are equivalent to the solution, being 1.

Thus, the correct selection is: All answers are correct.

Express the definition of root as a power:

$\sqrt[n]{a}=a^{\frac{1}{n}}$

Remember that in a square root (also called "root to the power of 2") we don't write the root's power as shown below:

$n=2$

Meaning:

$\sqrt{a}=\sqrt[2]{a}=a^{\frac{1}{2}}$

Now convert the roots in the problem using the root definition provided above. :

$\sqrt[6]{\sqrt{2}}=\sqrt[6]{2^{\frac{1}{2}}}=\big(2^{\frac{1}{2}}\big)^{\frac{1}{6}}$

In the first stage we applied the root definition as a power mentioned earlier to the inner expression (meaning inside the larger-outer root) and then we used parentheses and applied the same definition to the outer root.

Let's recall the power law for power of a power:

$(a^m)^n=a^{m\cdot n}$

Apply this law to the expression that we obtained in the last stage:

$\big(2^{\frac{1}{2}}\big)^{\frac{1}{6}}=2^{\frac{1}{2}\cdot\frac{1}{6}}=2^{\frac{1\cdot1}{2\cdot6}}=2^{\frac{1}{12}}$

In the first stage we applied the power law mentioned above and then proceeded first to simplify the resulting expression and then to perform the multiplication of fractions in the power exponent.

Let's summarize the various steps of the solution thus far:

$\sqrt[6]{\sqrt{2}}=\big(2^{\frac{1}{2}}\big)^{\frac{1}{6}} =2^{\frac{1}{12}}$

In the next stage we'll apply once again the root definition as a power, (that was mentioned at the beginning of the solution) however this time in the opposite direction:

$a^{\frac{1}{n}} = \sqrt[n]{a}$

Let's apply this law in order to present the expression we obtained in the last stage in root form:

$$$2^{\frac{1}{12}} =\sqrt[12]{2}$

We obtain the following result: :

$\sqrt[6]{\sqrt{2}}=2^{\frac{1}{12}} =\sqrt[12]{2}$

Therefore the correct answer is answer A.

In order to solve the given problem, we'll follow these steps:

• Step 1: Convert the inner square root to an exponent: $\sqrt{8} = 8^{1/2}$.

• Step 2: Apply the root of a root property: $\sqrt{\sqrt{8}} = (\sqrt{8})^{1/2} = (8^{1/2})^{1/2}$.

• Step 3: Simplify the expression using exponent rules: $(8^{1/2})^{1/2} = 8^{(1/2) \cdot (1/2)} = 8^{1/4}$.

The nested root expression simplifies to $8^{1/4}$.

Therefore, the simplified expression of $\sqrt{\sqrt{8}}$ is $8^{\frac{1}{4}}$.

After comparing this result with the multiple choice answers, choice 2 is correct.